Question

Assignment12: Problem 5 Previous Problem Problem List Next Problem (1 point) Find the general solution of the following system. Give the answer in real form (no complex numbers or complex exponentials) (t)(t)- 2y(t) C1 C2

Answer 1

Given dx/dt = x'(t) = -6x(t) - 5y(t) dy/dt = y'(t) = x(t) - 2y(t) now system can be written as (x'(t) y'(t)) = AX where x = (x(t) y(t)) A = [-6 1 -5 -2] Now Eigen values of A are given by [A - lambda I] = 0 implies lambda^2 + 8 lambda + 17 = 0 implies Lambda = -8 plusminus squareroot 64 - 68 /2 implies lambda = -8 plusminus 2c/2 implies lambda = -lambda + c, -lambda -c implies lambda_1 = -lambda + c, lambda_2 = -lambda -c Let X_1 be an eigen vector for lambda_1 = -lambda + c implies A X_1 = lambda_1 X_1 implies [-2 -c 1 -5 2 -c] [x y] = [0 0] implies [-2 -c 0 -5 0] [x y] = [0 0] implies (-2 -c)x - 5y = 0 implies (-2 -c)x = 5y implies x = 5/-2 -c y So Eigen vector I put y = 1 implies (x y) = (5/-2-c 1), lambda_1 = -y + c.....